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For some real number a and some positive integer n, the first few terms in the expansion of (1 + ax)^n are \[1 - 20x + 150x^2 + cx^3 + \dotsb.\]Find c.

 Mar 13, 2023
edited by Starziq  Mar 13, 2023
edited by Starziq  Mar 13, 2023
 #1
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We notice that the first three terms of the expansion are given as:

(1 + ax)^n = 1 + (n)(ax) + (n)(n-1)/(2!)(ax)^2 + ...

Comparing this with the given expansion, we have:

1 - 20x + 150x^2 + cx^3 + ... = 1 + (n)(ax) + (n)(n-1)/(2!)(ax)^2 + ...

Therefore, we can see that:

n = -20a

(n)(n-1)/(2!) = 150a^2

Solving for a and n, we have:

a = -3/10 and n = 12

Substituting these values back into the expansion, we have:

(1 - 0.3x)^12 = 1 - 20x + 150x^2 - 600x^3/7 + ...

Therefore, the missing coefficient is -600/7.

 Mar 13, 2023

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